Maxima Function
taylor (expr, x, a, n)
taylor(expr,[x_1,x_2,...],a,n)
taylor(expr,[x,a,n,'asymp])
taylor(expr,[x_1,x_2,...],[a_1,a_2,...],[n_1,n_2,...])
taylor(expr,[x_1,a_1,n_1],[x_2,a_2,n_2],...)
taylor (expr, x, a, n) expands the expression expr
in a truncated Taylor or Laurent series in the variable x
around the point a,
containing terms through (x - a)^n.
If expr is of the form f(x)/g(x)
and g(x) has no terms up to degree n
then taylor attempts to expand g(x) up to degree 2 n.
If there are still no nonzero terms, taylor doubles the
degree of the expansion of g(x)
so long as the degree of the expansion is less than or equal to n 2^taylordepth.
taylor (expr, [x_1, x_2, ...], a, n)
returns a truncated power series
of degree n in all variables x_1, x_2, ...
about the point (a, a, ...).
taylor (expr, [x_1, a_1, n_1], [x_2, a_2, n_2], ...)
returns a truncated power series in the variables x_1, x_2, ...
about the point (a_1, a_2, ...),
truncated at n_1, n_2, ....
taylor (expr, [x_1, x_2, ...], [a_1, a_2, ...], [n_1, n_2, ...])
returns a truncated power series in the variables x_1, x_2, ...
about the point (a_1, a_2, ...),
truncated at n_1, n_2, ....
taylor (expr, [x, a, n, 'asymp])
returns an expansion of expr in negative powers of x - a.
The highest order term is (x - a)^-n.
When maxtayorder is true, then during algebraic
manipulation of (truncated) Taylor series, taylor tries to retain
as many terms as are known to be correct.
When psexpand is true,
an extended rational function expression is displayed fully expanded.
The switch ratexpand has the same effect.
When psexpand is false,
a multivariate expression is displayed just as in the rational function package.
When psexpand is multi,
then terms with the same total degree in the variables are grouped together.
See also the switch for controlling expansion.
Examples:
(%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3); 2 2 (a + 1) x (a + 2 a + 1) x (%o1)/T/ 1 + --------- - ----------------- 2 8 3 2 3 (3 a + 9 a + 9 a - 1) x + -------------------------- + . . . 48 (%i2) %^2; 3 x (%o2)/T/ 1 + (a + 1) x - -- + . . . 6 (%i3) taylor (sqrt (x + 1), x, 0, 5); 2 3 4 5 x x x 5 x 7 x (%o3)/T/ 1 + - - -- + -- - ---- + ---- + . . . 2 8 16 128 256 (%i4) %^2; (%o4)/T/ 1 + x + . . . (%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2); inf /===\ ! ! i 2.5 ! ! (x + 1) ! ! i = 1 (%o5) ----------------- 2 x + 1 (%i6) ev (taylor(%, x, 0, 3), keepfloat); 2 3 (%o6)/T/ 1 + 2.5 x + 3.375 x + 6.5625 x + . . . (%i7) taylor (1/log (x + 1), x, 0, 3); 2 3 1 1 x x 19 x (%o7)/T/ - + - - -- + -- - ----- + . . . x 2 12 24 720 (%i8) taylor (cos(x) - sec(x), x, 0, 5); 4 2 x (%o8)/T/ - x - -- + . . . 6 (%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5); (%o9)/T/ 0 + . . . (%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5); 2 4 1 1 11 347 6767 x 15377 x (%o10)/T/ - -- + ---- + ------ - ----- - ------- - -------- 6 4 2 15120 604800 7983360 x 2 x 120 x + . . . (%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6); 2 2 4 2 4 k x (3 k - 4 k ) x (%o11)/T/ 1 - ----- - ---------------- 2 24 6 4 2 6 (45 k - 60 k + 16 k ) x - -------------------------- + . . . 720 (%i12) taylor ((x + 1)^n, x, 0, 4); 2 2 3 2 3 (n - n) x (n - 3 n + 2 n) x (%o12)/T/ 1 + n x + ----------- + -------------------- 2 6 4 3 2 4 (n - 6 n + 11 n - 6 n) x + ---------------------------- + . . . 24 (%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3); 3 2 y y (%o13)/T/ y - -- + . . . + (1 - -- + . . .) x 6 2 3 2 y y 2 1 y 3 + (- - + -- + . . .) x + (- - + -- + . . .) x + . . . 2 12 6 12 (%i14) taylor (sin (y + x), [x, y], 0, 3); 3 2 2 3 x + 3 y x + 3 y x + y (%o14)/T/ y + x - ------------------------- + . . . 6 (%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3); 1 y 1 1 1 2 (%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x y 6 2 6 3 y y 1 3 + (- -- + . . .) x + . . . 4 y (%i16) taylor (1/sin (y + x), [x, y], 0, 3); 3 2 2 3 1 x + y 7 x + 21 y x + 21 y x + 7 y (%o16)/T/ ----- + ----- + ------------------------------- + . . . x + y 6 360